ANOVA Calculator
One-way analysis of variance from per-group summaries: is the spread BETWEEN group means larger than the spread INSIDE the groups? The F ratio, both mean squares, the partition proved live, and the p from the incomplete beta.
ANOVA Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- One-way, fixed-effects, from summaries — no repeated measures
- Post-hoc pairwise attribution is out of scope by design
In short: Three groups — 5, 62, 4.2; 5, 68, 3.9; 5, 74, 4.6 — give a grand mean of 68, SSB = 5×(36 + 0 + 36) = 360 between and SSW = 4×(17.64 + 15.21 + 21.16) = 216.04 within, and the partition SST = 576.04 closes exactly. MSB = 180 on df 2, MSW = 18.003333 on df 12, so F = 9.998148 and p = 0.002783: the between-group spread runs about ten times the within-group spread, and the equal-means claim does not survive. Which group differs from which is a separate question — ANOVA convicts the collection, not a culprit.
Formula
SSB = Σnᵢ(x̄ᵢ − x̄)² · SSW = Σ(nᵢ−1)sᵢ² · F = (SSB/(k−1)) / (SSW/(N−k))
The partition SST = SSB + SSW is an identity, and the page proves it on your numbers instead of asserting it. p = P(F > F) from the regularized incomplete beta.
Worked Example
- Enter one group per ‘;’ as n, mean, s.
- Check the groups card echoes your summaries.
- Read F, the partition with the identity closing exactly, and p.
- Compare p to your bar; then ask the follow-up question ANOVA refuses to answer.
5, 62, 4.2; 5, 68, 3.9; 5, 74, 4.6: SSB 360, SSW 216.04, SST 576.04, MSB 180, MSW 18.003333, F = 9.998148 on df 2 and 12, p = 0.002783. With k = 2 the same F equals the squared pooled t — the identities agree.
Strengths & Limits Of This Model
Where this engine is strong
- The variance partition computed and closed to the cent
- Group summaries echoed so bad input is visible
Where it stops
- No Welch correction for unequal variances
- No effect size (η²) — deferred to the effect-size page
Practical Use Cases
Experiment triage
three arms, one verdict on the set
Process lines
do the lines share a mean?
Teaching
the partition and F built from summaries you can check by hand
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
ANOVA Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why compare spreads to ask about means?
Because that IS the comparison: if all the group means were one common value, the variation between them would be sampling noise of the same size as the variation within. F is the ratio of those two spread estimates, and “means differ” is exactly “between outgrows within.”
Why does ANOVA refuse to name the winning group?
Because rejecting equal means is a verdict about the COLLECTION. Picking the biggest mean afterwards and calling it the winner is the multiple-comparisons trap: k groups make k(k−1)/2 pairwise comparisons, and the bar must rise with the count. Pairwise tools with corrections live elsewhere; this page convicts no individual.
What do the summaries assume?
Roughly normal residuals, independent observations, and comparable group variances — the s values you enter are printed beside each other precisely so heteroscedasticity is visible before you trust F. If one group’s s is triple another’s, read the p with suspicion.
Why does a zero s in every group get refused?
Because MSW would be zero and F would grow without bound — the test degenerates. Real data always wobble; a summary set with no within-group spread anywhere is telling you the summary is wrong, and the page believes the data over the summary.
Does unequal group size break anything?
No — the formulas carry each group’s own n. Balance only affects power and robustness: unequal n with unequal variances couples the two problems, which is why the classic advice is to balance when you can and check the s values when you cannot.
Is this related to the t test?
Directly: with two groups the F here equals the squared pooled t, and the p-values agree exactly. ANOVA is the generalization of that one comparison to k groups — the verifier checks the identity numerically on the two-group case.
What is a mean square, physically?
A variance estimate: a sum of squares divided by its degrees of freedom. MSB estimates how much group means wander if they share one true mean; MSW estimates the pure within-group noise. F is the ratio — near 1 if the means agree, far above 1 if the groups genuinely differ.
Can a tiny difference between means reach significance?
Yes, with enough n — F prices sample size in, so trivial gaps become detectable. That is why the verdict card reports the RATIO rather than letting the p alone pose as an effect size; sizing the gap honestly is a separate question with a separate tool.